The goal of this paper is to demonstrate the convergence of circle packings to uniformization maps in an alternative framework proposed in Gu-Luo-Wu [1]} and Wu-Zhu [2]} when the surface is a smooth torus. This scheme of convergence is proposed for a new theory of discrete conformal geometry called vertex scaling. The fundamental results about this new theory include a discrete uniformization theorem for polyhedral surfaces [3]}, [4]}, [5]}, an infinite rigidity theorem similar to the hexagonal circle packings in the plane [6]}, and the convergence of discrete conformal maps to their smooth counterparts[7]}. Effective algorithms are also available to compute conformal maps [8]}.
